## Propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous Maxwellian gas.(English)Zbl 0927.76088

Summary: We prove an inequality for the gain term in the Boltzmann for equation for Maxwellian molecules that implies a uniform bound on Sobolev norms of the solution, provided the initial data has a finite norm in the corresponding Sobolev space. Then we prove a sharp bound on the rate of exponential convergence to equilibrium in a weak norm. These results are combined, using interpolation inequalities, to obtain the optimal rate of exponential convergence in the strong $$L^1$$ norm and in various Sobolev norms. These results are the first showing that the spectral gap in the linearized collision operator actually does govern the rate of approach to equilibrium for the full nonlinear Boltzmann equation, even for initial data that is far from equilibrium.

### MSC:

 76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics 82C40 Kinetic theory of gases in time-dependent statistical mechanics
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